Fast rotational relaxation — When \(l_{\theta } << \delta \phi R\) , as well as, \(\tau _{\theta }\)  is much shorter than any other relevant time scale of the system, it can be assumed that self-propellers exhibit almost uncorrelated diffusive motion inside the cavity. The effective diffusion constant of the active particle, \(D_{eff}=D_0 + v_0^2/2D_\theta \) , carries the contribution from both thermal motion as well as the self-propulsion. In this limit, the success rate in random search of the opening of the diffusive particle is directly proportional to its diffusivity. Following [1]} the exit time can be expressed as,
\(\tau _{ex} = \frac{R^2}{D_0+l_\theta v_0/2}\left[\ln \left(\frac{2\pi -\delta \phi }{\delta \phi }\right)+\chi \right], \) 
